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Mat. Sb., 2017, Volume 208, Number 9, Pages 148–170 (Mi msb8880)  

This article is cited in 3 scientific papers (total in 3 papers)

Minimal cubic surfaces over finite fields

S. Yu. Rybakov, A. S. Trepalin

Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute), Moscow

Abstract: Let $X$ be a minimal cubic surface over a finite field $\mathbb{F}_q$. The image $\Gamma$ of the Galois group $\operatorname{Gal}(\overline{\mathbb{F}}_q / \mathbb{F}_q)$ in the group $\operatorname{Aut}(\operatorname{Pic}(\overline{X}))$ is a cyclic subgroup of the Weyl group $W(E_6)$. There are $25$ conjugacy classes of cyclic subgroups in $W(E_6)$, and five of them correspond to minimal cubic surfaces. It is natural to ask which conjugacy classes come from minimal cubic surfaces over a given finite field. In this paper we give a partial answer to this question and present many explicit examples.
Bibliography: 11 titles.

Keywords: finite field, cubic surface, zeta function, del Pezzo surface.

Funding Agency Grant Number
Russian Science Foundation 14-50-00150
This research was carried out at the Institute for Information Transmission Problems of the Russian Academy of Sciences with the financial support of the Russian Science Foundation (project no. 14-50-00150).


DOI: https://doi.org/10.4213/sm8880

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English version:
Sbornik: Mathematics, 2017, 208:9, 1399–1419

Bibliographic databases:

Document Type: Article
UDC: 512.774.7
MSC: Primary 11G25; Secondary 14J20
Received: 12.12.2016 and 05.04.2017

Citation: S. Yu. Rybakov, A. S. Trepalin, “Minimal cubic surfaces over finite fields”, Mat. Sb., 208:9 (2017), 148–170; Sb. Math., 208:9 (2017), 1399–1419

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. Trepalin, “Minimal del Pezzo surfaces of degree 2 over finite fields”, Bull. Korean. Math. Soc., 54:5 (2017), 1779–1801  crossref  mathscinet  isi  scopus
    2. J. Little, H. Schenck, “Codes from surfaces with small Picard number”, SIAM J. Appl. Algebr. Geom., 2:2 (2018), 242–258  crossref  mathscinet  zmath  isi
    3. S. G. Vlăduţ, D. Yu. Nogin, M. A. Tsfasman, “Varieties over finite fields: quantitative theory”, Russian Math. Surveys, 73:2 (2018), 261–322  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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